Covariance structures of Gaussian and log-Gaussian vector stochastic processes

Abstract

Although the covariance structures of univariate Gaussian and log Gaussian stochastic processes have been extensively studied in the past few decades, the development of covariance structures for Gaussian and log-Gaussian vector stochastic processes is still in the early stages. Speci cally, there has been little discussion about how to construct the covariance matrix functions of multivariate Gaussian time series with long memory, especially ones with power-law and log-law decaying covariance structures. Furthermore, there have been relatively few results about how to determine whether a given matrix function is the covariance matrix function of a log-Gaussian vector random field. This dissertation provides new methods for identifying and constructing covariance matrix functions of Gaussian vector time series and log-Gaussian vector random fields. In particular, research is presented on how to fi nd covariance matrix structures with power-law decaying and log-law decaying direct and cross covariances. Also, the intricate relationship between the mean function and the covariance matrix function of the log-Gaussian vector random fi eld is explored. In additon, operation preserving properties are investigated for the log-Gaussian vector random field.